Patient-Specific Models Take Aim at Uncertainty in Medical Treatment

By Barry A. Cipra

SIAM News, October 21, 2005

Picture yourself in a hospital bed, about to go under the knife. Your doctor stops by to explain the surgery he thinks is your best bet to correct that pesky heart problem you created with your cheeseburger-and-french-fry diet. He shows you the surgical plan, a hand-scribbled cartoon of a descending aorta, with a grafted-on bypass in the shape of the letter C. Who, you wonder, was this guy's last patient? Homer Simpson?

Not to worry. The surgeon reassures you that he's done this operation hundreds of times, and it corrects your kind of condition in nearly eighty percent of cases. Which leaves you with a big question: Could we be a little more certain I'm not one of the other twenty percent?

Charles Taylor thinks we can be a lot more certain. He and colleagues at Stanford University's Cardiovascular Biomechanics Research Lab are working toward a future in which physicians use computer-based imaging and monitoring not just to diagnose their patients' ailments, but also for patient-specific simulations to test alternative treatments. Taylor gave a talk on predictive medicine and its mathematical underpinnings in July at the SIAM Annual Meeting in New Orleans.

Medical diagnostics has had a quantitative aspect since the first nurse took the first patient's temperature. Computers entered the mix with the advent of CAT scans and MRI. Surgeons nowadays rely on computational methods to help them visualize their patients' innards before they open them up, and even to perform parts of operations. Yet one crucial step is still mainly qualitative: Doctors recommend treatments largely on the basis of the profession's empirical experience, comparing the patient at hand with the "typical" patient who has the same condition.

Taylor contrasts this with the situation in engineering. Medical imaging and robotic surgery can be likened to computer-assisted design and manufacture. But computer-assisted engineering goes far beyond CAD–CAM. Engineers use computer simulations of the systems they design and build to test how they'll behave in the real world and to explore what-if alternatives in design or construction. The aerodynamic properties of a fuselage, for example, or the stability of a high-rise building on the San Andreas fault, are calculated from detailed models of specific designs, not merely guessed at from experience with similar designs. Back-of-the-envelope estimates remain a venerable tradition, of course, but they no longer serve as the final word in justifying multi-million-dollar investments.

Predictive medicine aims to provide the same sort of high-tech precision and specificity for sick people. The goal is to convert medical imaging and monitoring data into a model of an individual's anatomy and physiology, and then use the model to predict the patient's response to alternative treatments, be they surgical or pharmacological, under different physiological conditions. Achieving this goal will require the same sort of mathematical and computational tools that have revolutionized modern engineering.

Taylor's team has focused on the cardiovascular system, in part because of its prominence in mortality statistics and in part because it lends itself to biomechanical study. Blood flow is described with variables a hydrologist can recognize: pressure, resistance, and flow rate, among others. The basic equations for the plumbing in your body are the same as those for the plumbing in your house. All you need to compute blood flow is a good grasp of cardiovascular geometry (anatomy) and a handle on boundary and forcing conditions (physiology).

That's where it gets tricky, of course. The fractaline branching of the cardiovascular system is far more complicated than the plumbing in a house--or even the water/sewer system of a metropolis like New York. (This is one reason surgeons are paid even more than plumbers. A ruptured aneurysm is not unlike a leaky pipe, but you can't home-repair an aneurysm with duct tape.) Blood vessels are also elastic in ways that copper and cast iron aren't. The pulsatile nature of blood flow adds to the complexity. Finally, as extensively monitored as patients are these days, physicians don't have access to all the data they necessarily need.

Nevertheless, Taylor says, computer simulations can give quantitative insight into what will happen when, for example, a surgeon bypasses an occluded artery (see Figure 1). Image segmentation techniques, in conjunction with a method like level sets, turn MRI data into a geometric model, which the surgeon can manipulate to simulate postoperative conditions. Finite element methods discretize the geometric model and solve the equations of blood flow, using physiological data from the patient to simulate conditions of rest or exercise. Computer visualization methods present the results in an interactive format.

As they bring patient-specific detail to their computer models, Taylor and colleagues are also exploring analytic ways to simplify the calculations. In particular, it can take hours, even on high-performance parallel computers, to solve the fully three-dimensional, time-dependent equations for blood flow. The effort is warranted for some details, such as flow recirculation and shear stress on artery walls. But it is not strictly necessary for determining one-dimensional downstream data, such as mean flow rates and pressure losses. Taylor's group has studied a one-dimensional finite element method that returns such relevant information after just minutes of computation on a PC.

To date, no actual clinical case has been decided on the basis of predictive simulations, but Taylor says that he and colleagues have had success with a number of retrospective studies--i.e., their models have shown good agreement with postoperative results in a number of subjects. The researchers have been refining their models via in vivo animal experiments. In one study, they compared calculated and measured flow values for a bypass procedure performed in eight pigs. (Cardiovascularly, pigs and humans have a lot in common.) The results (see Figures 2 and 3) suggest that predictive medicine may someday play an important role in helping physicians decide how they can best treat their patients.

Barry A. Cipra is a mathematician and writer based in Northfield, Minnesota.

Figure 1. Curvature-based mesh refinement leads to a finite element mesh that captures the relevant geometry of a proposed bypass in a mock clinical case of a patient with multiple occlusions in the iliac and femoral arteries. Reprinted with permission from C.A. Taylor et al., "Predictive Medicine: Compu-tational Techniques in Therapeutic Decision-Making," Computer Aided Surgery, 4 (1999), 231–247.

	Figure 2. Actual and simulated mean flow rates, before and after surgery, in three of eight experimental pigs. The shorter pair of bars for each pig represents preoperative values, actual (left) and simulated (right); the taller pair represents postoperative values.

	Figure 3. Actual and simulated flow rates over one cardiac cycle for the third pig in Figure 2 show similar waveforms (the solid curve shows measured inlet flow). Re-printed with permission from J.P. Ku et al., "In Vivo Validation of Numerical Prediction of Blood Flow in Arterial Bypass Grafts," Annals of Biomedical Engineering, 30 (2002), 743–752.